Question

Помогите решить уравнение

ctg(x+pi/4)=tg(2x-pi/4)

Answer 1

$ctg(x+\frac{\pi }{4})=tg(2x-\frac{\pi}{4})\\\\ODZ:\; \; sin(x+\frac{\pi}{4})\ne 0\; ,\; cos(2x-\frac{\pi}{4})\ne 0\; ,\\\\\star \; \; ctga=tg(\frac{\pi}{2}-a)\; \; \to \; \; ctg(x+\frac{\pi}{4})=tg(\frac{\pi}{2}-x-\frac{\pi}{4})=tg(\frac{\pi}{4}-x)\; \star \\\\tg(\frac{\pi}{4}-x)=tg(2x-\frac{\pi}{4})\; \; ,\; \; \; \; tg\frac{\pi}{4}=1\\\\\frac{1-tgx}{1+1\cdot tgx}=\frac{tg2x-1}{1+1\cdot tg2x}\\\\\frac{1-tgx}{1+tgx}=\frac{tg2x-1}{1+tg2x}\\\\\frac{(1-tgx)(1+tg2x)-(tg2x-1)(1+tgx)}{(1+tgx)(1+tg2x)}=0$

$\frac{2-2tgx\vfot tg2x}{(1+tgx)(1+tg2x)}=0\; \; \to \; \; \frac{2\cdot (1-tgx\cdot tg2x)}{(1+tgx)(1+tg2x)}=0\; \; \to \; \; \left \{ {{tgx\cdot tg2x=1} \atop {1+tgx\ne 0;\; 1+tg2x\ne 0}} \right.\\\\\frac{sinx\cdot sin2x}{cosx\cdot cos2x}=1\; \; \to \; \; \frac{sinx\cdot sin2x}{cosx\cdot cos2x}-1=0\; ,\; \; \frac{sinx\cdot sin2x-cosx\cdot cos2x}{cosx\cdot cos2x} =0\; ,\\\\\frac{-cos(x+2x)}{cosx\cdot cos2x}=0\; ,\; \; \frac{cos3x}{cosx\cdot cos2x}=0\; \; \to \; \; \left \{ {{cos3x=0} \atop {cosx0\; ,\; cos2x\ne 0ne }} \right.$

$cos3x=0\; ,\; 3x=\frac{\pi}{2}+\pi n\; \; ,\; \; x=\frac{\pi}{6}+\frac{\pi n}{3}\; ,\; n\in Z\\\\cosx\ne 0\; ,\; \; x\ne \frac{\pi}{2}+\pi k\; ,\; k\in Z\\\\cos2x\ne 0\; ,\; \; x\ne \frac{\pi}{4}+\frac{\pi m}{2}\; ,\; m\in Z\\\\tgx\ne -1\; ,\; \; x\ne -\frac{\pi}{4}+\pi l\; ,\; l\in Z\\\\tg2x\ne -1\; ,\; \; x\ne -\frac{\pi}{8}+\frac{\pi t}{2}\; ,\; t\in Z\\\\Otvet:\; \; x=\frac{\pi}{6}+\pi n\; ,\; \; x=\frac{5\pi }{6}+\pi n\; ,\; n\in Z\; .$